The glassy state is ubiquitous in nature and technology. The most common way of making a glass is by cooling a liquid sufficiently fast so that it does not have time to crystallize. The manner in which such supercooled liquids acquire amorphous rigidity is poorly understood. This lack of knowledge impacts negatively on the design, formulation, and manufacturing of important products in the pharmaceutical, food, communications, energy, and engineering plastics industries. We review important recent advances in the fundamental understanding of glasses that have resulted from two complementary statistical mechanical viewpoints: the energy landscape formalism and statistical geometry. The former provides a unifying analytical framework for describing the thermodynamic and transport properties of glasses and the viscous liquids from which they are commonly formed. Statistical geometry addresses the quantitative description of a glassy material's history-dependent structure.